The angle that holds the lock

The previous essay ended with a face that does nothing to the wheel at all, and left something out. A locking face cut exactly as an arc about the pallet arbor is not merely free of recoil. It is also free of any tendency to stay locked, and the two facts are the same fact.

A lock that is standing on its point #

Ask what the resting tooth does to the pallet. It pushes along the face’s normal, and for an arc centred on the arbor that normal is radial: it points straight at the pivot. A force through a pivot has no moment about it.

So the pallet, with a tooth pressed against it and the whole train pushing, feels nothing turning it either way. The lock is in equilibrium and the equilibrium is indifferent — a pencil balanced on its point rather than a ball in a bowl. Any disturbance whatever, a knock on the case or the pendulum’s own return, moves the pallet, and nothing brings it back.

The number is $6.5 \times 10^{-17}$ per unit of wheel torque, which is zero, at every depth of lock tested from 0.2° to 2.5°. Not small at some depths and larger at others: zero throughout, because the geometric reason for it does not care where along the arc the tooth sits.

Tilting the face #

The remedy is to tilt the locking face by a small angle — in horology, to give it draw. Rotate the arc about the corner by $\delta$ and the normal at the contact rotates with it, so it no longer aims at the arbor, and its moment arm about the arbor becomes

$$\rho \sin \delta$$

where $\rho$ is the pallet arm. On the escapement this field is built around, $\rho = 0.8588$ wheel radii, so a tilt of 1.5° gives an arm of $0.022481$ — and the arm computed from the geometry at the corner is $0.022482$. The two agree to twelve figures, which is the point of computing it at all: the closed form is exact only at the corner, where the face’s own radius is exactly the pallet arm, and at the depth the tooth actually rests it is out by five parts in a hundred thousand.

The sign of that moment can go either way, and which way it goes is not a convention. It is asked: build the escapement with the tilt one way and the tooth’s push carries the pallet deeper into the lock; build it the other way and the push lifts the pallet out, and the escapement releases itself as soon as it locks. The library computes which and refuses the second.

The identity #

Here is the part that makes this a single quantity rather than two.

Nothing is stored in a rigid mechanism and nothing is lost at a frictionless contact, so over any small motion the work the wheel’s torque does and the work done on the pallet must cancel:

$$\tau \, \mathrm{d}\theta + M \, \mathrm{d}\varphi = 0 \quad \Longrightarrow \quad \frac{M}{\tau} = -\frac{\mathrm{d}\theta}{\mathrm{d}\varphi}$$

The left-hand side is the draw, expressed as the pallet’s moment per unit of wheel torque. The right-hand side is minus the recoil rate: how fast the wheel is driven backwards as the pallet goes deeper.

They are measured separately here — the draw from a cross product of a contact normal with a lever arm, the recoil rate by differencing the contact solve at two nearby pallet angles — and they agree to nine figures at every tilt tested. At 1.5° of tilt both are $0.0266669$.

So: an escapement cannot be given a lock that holds itself without also being given a wheel that is pushed backwards, and the amount of the one is the amount of the other. There is no design that separates them, because they are not two properties of the mechanism. They are one number, read once as a moment and once as a rate.

That also settles the question the previous essay left open. A deadbeat escapement is not a mechanism with no recoil; it is a mechanism whose recoil is exactly as large as its draw, and its draw has to be large enough to hold. The zero-recoil escapement exists — it is the one in the first figure — and nobody builds it because it does not stay locked.

How much draw is used #

The two ends of the range in practice are about a factor of seven apart, and the reason for the difference is what the lock has to survive.

A clock’s deadbeat carries 1.5° to 2°. Its pallets are attached to a pendulum that swings slowly and predictably, nothing much disturbs the case, and the lock has only to resist the pendulum’s own return. At 1.5° the mechanism recoils 0.106° at a 4° supplementary arc, which is 1.8% of the wheel’s travel for the beat.

A watch’s lever escapement carries 10° to 15°, and the reason is not that its lock is harder to hold. It is that the lever escapement’s lock is doing a second job: holding the lever hard against its banking pin while the balance is away, so that a shock cannot throw the lever across and unlock the wheel at the wrong moment. That is the safety action, and it is the next essay.

And that is the resolution of what would otherwise be a contradiction. Recoil is expensive when the thing recoiling is attached to the oscillator, because the wheel is then pushing back on the pendulum through the whole supplementary arc. In a detached escapement the pallets are on a lever that stands still against a banking pin during the lock, so the recoil happens once as the tooth settles and then nothing moves at all. The same 10° of draw that would ruin a clock costs a watch nothing, because a watch’s lock does not move.

The flat face’s accidental draw #

The flat face has draw without anybody having given it any, and the amount is not negligible.

A flat cut tangent to the concentric arc has no draw at the corner — the tangent’s normal points at the arbor, exactly as the arc’s does. A degree below the corner it does not, because the flat has departed from the arc and the departure has a direction. Measured as a moment per unit of wheel torque against the depth of lock, the flat gives 0.0053 at 0.3°, 0.0106 at 0.6°, 0.0214 at 1.2° and 0.0437 at 2.4°: a straight line through the origin, with a slope such that the accidental draw at a lock of $\lambda$ is very nearly the deliberate draw of an arc tilted by $\lambda$.

At a lock of 1.2° the flat’s draw is 0.0214, against 0.0267 for an arc deliberately tilted 1.5°. So a flat-faced escapement locking 1.2° deep behaves, to within a fifth, like an arc-faced escapement drawn at 1.2° — and its draw goes up and down with its lock, which the arc’s does not.

That is the practical case against flats, stated without reference to recoil at all. An escapement whose draw is a consequence of its lock depth is an escapement whose most important safety property changes when the depth of lock is adjusted, and the depth of lock is the one thing a repairer routinely adjusts. Deepen the lock to stop it tripping and the draw increases; shallow it to reduce the unlocking effort and the draw goes with it, in proportion, unasked.

An arc-faced escapement has a draw that is set once, by the angle its faces were struck at, and is untouched by every subsequent adjustment. That is a different kind of design and it is why the substitution mattered.

What a zero moment arm is not #

It is worth being careful about one thing, because the word “zero” has done a lot of work in this essay and the site has a rule about it.

The moment at $\delta = 0$ is $6.5 \times 10^{-17}$, which is not zero — it is the rounding error left over from computing a cross product of two vectors that are parallel to a dozen significant figures. The claim being made is not that the arithmetic returned an exact zero. It is that the geometry has a zero there, that the arithmetic agrees to the precision available, and that the quantity is identically zero at every depth of lock rather than accidentally small at one.

The distinction matters because the alternative reading — a very small draw rather than none — would suggest a mechanism that mostly holds and occasionally does not, and that is not what an indifferent equilibrium is. A pencil balanced on its point is not a pencil that is nearly standing up.

The same test, for the third time #

This is now the third mechanism in this field decided by one cross product, and the family resemblance is the argument rather than an observation.

What the three have in common is not a technique. It is that a contact — a pin, a face, a tooth — can only transmit along one direction, and once that direction is known the sense of everything downstream is settled by geometry. Magnitudes need forces. Senses do not.

Unlocking, which the draw also decides #

Draw has a third consequence and it is the one a watchmaker meets first: it decides how hard the escapement is to unlock.

The tooth is resting on a tilted face, and to release it the pallet has to be turned against the moment the tilt produces — the same moment, the same lever arm, working the other way. So an escapement with more draw is an escapement that takes more to unlock, and takes it from the oscillator, which is the one place a designer least wants to take anything from.

The geometry of that is a run: the pallet has to travel from where the tooth landed back to the corner before the impulse can begin, and over that travel the wheel is being driven backwards the whole way. It is the third term of the beat budget — the lock-in run — seen from the other end, and it is exactly zero on the untilted arc and exactly $-0.032°$ on the arc drawn at 1.5°.

So the same angle appears for the fourth time: as a moment that holds the lock, as a rate at which the wheel recoils, as a run in the wheel’s budget, and as an effort the oscillator has to supply to get free. Four quantities in four different units, one geometric parameter, and no two of them measured the same way.

The last of the four is where the design pressure comes from. A clock’s pendulum has plenty to spare and takes 1.5°; a watch’s balance has very little and would prefer none, and takes 10° anyway because the alternative is a lever that can be shaken out of its banking. The whole of escapement design is that trade, and this essay is the reason it is a trade and not two independent choices.

The identity is pointwise, which explains the flat’s curve #

The virtual-work argument equates two quantities at a configuration, and reading it as a statement about whole faces rather than about points is what makes the previous essay’s tables look mysterious. Taken pointwise it explains them.

A tilted arc has the same draw wherever the tooth rests on it: the face is concentric, the tilt is uniform, and the moment arm is the same at every depth. Its recoil rate is therefore the same at every depth too, and the recoil accumulated over a supplementary arc is that constant rate times the arc — linear. Which is exactly what the previous essay measured: 0.0351°, 0.0706°, 0.1064°, 0.1427° for tilts of 0.5°, 1°, 1.5° and 2°, in proportion to the tilt, and linear in the arc as well.

A flat has draw that grows as the tooth rests further below the corner, because the flat departs from the concentric arc more the further along it one goes. So its recoil rate grows with depth, and the recoil accumulated over an arc is the integral of a growing rate rather than a constant one times a length. That is why the flat’s recoil at a lock of 1.2° came out as 0.0130°, 0.0306°, 0.0800°, 0.2381° and 0.8120° with successive ratios climbing from 2.36 towards 4 — a linear term on top of a quadratic one, which is precisely what integrating a rate with a linear term in it produces.

So the two essays’ tables are one table. The draw column here and the recoil column there are the same quantity measured at the same configurations by two entirely different routes — one a cross product of a contact normal with a lever arm, the other a swept displacement of the escape wheel — and their agreement is a check of unusual strength, because a mistake in either would have to be reproduced exactly in the other to escape notice.

It also gives a design consequence the constant-draw case hides. A flat’s draw is a function of how deeply it is locked, so an escapement with flat faces has a self-holding tendency that varies with the depth of lock, and depth of lock varies with everything — with the banking, with wear, with how far the pallets have been let into the wheel. An arc’s draw is set once, by the angle its faces were struck at, and no adjustment of depth changes it. That is the same argument the previous essay made about recoil, and it had to be, because it is the same angle.

The degenerate case is the interesting one #

One last observation, because it puts this field’s central object in the same category as the site’s oldest one.

At $\delta = 0$ the draw vanishes and the escapement stops being able to hold itself. At a four-bar’s toggle the transmission angle vanishes and the linkage stops being able to drive its output. At a parallel platform’s direct singularity the constraint wrenches become dependent and the platform stops being controllable. In every case the mechanism has not run out of configurations, and its Jacobian in the ordinary sense is perfectly healthy — what has happened is that a moment arm has gone to zero.

The difference here is which side of it a designer wants to be on. A toggle is avoided; a singularity is avoided; a lock with no draw is avoided. But an escapement is the one mechanism of the three that is deliberately placed close to its degeneracy, because the whole design problem is to have just enough draw to hold and no more. One and a half degrees is what “just enough” turned out to mean, and what it costs is a tenth of a degree of wheel motion, twice a second, for as long as the clock runs.

That is the last of the four readings, and it is worth putting the number to it. A seconds pendulum beats 86,400 times a day, and a tenth of a degree of recoil at each beat is 8,640° of extra wheel travel — twenty-four full turns of the escape wheel, every day, going backwards and forwards through the same tenth of a degree of the same thirty tooth tips. Nothing in that calculation is dynamics; it is a length multiplied by a count. What it explains is why the working faces of a deadbeat’s pallets are the parts of a clock most likely to be found polished into hollows, and why a mechanism whose whole virtue is that its wheel does not move turns out to move its wheel more than any other part of the clock.

One practical reading of the pointwise identity, since it turns two measurements into one. An escapement’s draw can be measured statically — hold the wheel, press the pallet, and see which way it tends — and its recoil can only be measured by running the mechanism and watching the wheel. The identity says the first measurement predicts the second, at every depth of lock, so a bench check that takes a moment stands in for a sweep that does not. That is worth having in a subject where the mechanism under test is usually inside a watch, and it is the sort of consequence a conservation argument produces and a pair of independent computations does not.